Obviously there exist infinite ways to purify \(\rho _{AB}\) . The EoP is defined by minimizing the entanglement entropy \(S_{AA^{\prime }}\)  over all purifications of \(\rho _{AB}\)  [1]}

Ep(AB)=AA'BB'min(SAA'),

where \(S_{AA^{\prime }}\)  is the entanglement entropy corresponding to the density matrix \(\rho _{AA^{\prime }}\)  and \(\rho _{AA^{\prime }}={\rm {Tr}}_{BB^{\prime }}\big [\left(|\psi \rangle _{ABA^{\prime }B^{\prime }}\right) \left({}_{ABA^{\prime }B^{\prime }}\langle \psi |\right)\big ]\) .
